3.653 \(\int x^4 (A+B x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=114 \[ \frac{x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{6 (a+b x)}+\frac{a A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b B x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]

[Out]

(a*A*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + ((A*b + a*B)*x^6*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(6*(a + b*x)) + (b*B*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(7*(a + b*x))

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Rubi [A]  time = 0.224553, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{6 (a+b x)}+\frac{a A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b B x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(a*A*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + ((A*b + a*B)*x^6*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(6*(a + b*x)) + (b*B*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(7*(a + b*x))

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Rubi in Sympy [A]  time = 20.4602, size = 117, normalized size = 1.03 \[ \frac{B x^{5} \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{14 b} + \frac{a x^{5} \left (7 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{210 b \left (a + b x\right )} + \frac{x^{5} \left (7 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{42 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)*((b*x+a)**2)**(1/2),x)

[Out]

B*x**5*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(14*b) + a*x**5*(7*A*b - 5
*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(210*b*(a + b*x)) + x**5*(7*A*b - 5*B*a)*
sqrt(a**2 + 2*a*b*x + b**2*x**2)/(42*b)

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Mathematica [A]  time = 0.0395288, size = 49, normalized size = 0.43 \[ \frac{x^5 \sqrt{(a+b x)^2} (7 a (6 A+5 B x)+5 b x (7 A+6 B x))}{210 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x^5*Sqrt[(a + b*x)^2]*(7*a*(6*A + 5*B*x) + 5*b*x*(7*A + 6*B*x)))/(210*(a + b*x)
)

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Maple [A]  time = 0.008, size = 44, normalized size = 0.4 \[{\frac{{x}^{5} \left ( 30\,Bb{x}^{2}+35\,Abx+35\,aBx+42\,aA \right ) }{210\,bx+210\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)*((b*x+a)^2)^(1/2),x)

[Out]

1/210*x^5*(30*B*b*x^2+35*A*b*x+35*B*a*x+42*A*a)*((b*x+a)^2)^(1/2)/(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267328, size = 36, normalized size = 0.32 \[ \frac{1}{7} \, B b x^{7} + \frac{1}{5} \, A a x^{5} + \frac{1}{6} \,{\left (B a + A b\right )} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*x^4,x, algorithm="fricas")

[Out]

1/7*B*b*x^7 + 1/5*A*a*x^5 + 1/6*(B*a + A*b)*x^6

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Sympy [A]  time = 0.197228, size = 29, normalized size = 0.25 \[ \frac{A a x^{5}}{5} + \frac{B b x^{7}}{7} + x^{6} \left (\frac{A b}{6} + \frac{B a}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)*((b*x+a)**2)**(1/2),x)

[Out]

A*a*x**5/5 + B*b*x**7/7 + x**6*(A*b/6 + B*a/6)

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GIAC/XCAS [A]  time = 0.271428, size = 105, normalized size = 0.92 \[ \frac{1}{7} \, B b x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, B a x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, A b x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, A a x^{5}{\rm sign}\left (b x + a\right ) - \frac{{\left (5 \, B a^{7} - 7 \, A a^{6} b\right )}{\rm sign}\left (b x + a\right )}{210 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*x^4,x, algorithm="giac")

[Out]

1/7*B*b*x^7*sign(b*x + a) + 1/6*B*a*x^6*sign(b*x + a) + 1/6*A*b*x^6*sign(b*x + a
) + 1/5*A*a*x^5*sign(b*x + a) - 1/210*(5*B*a^7 - 7*A*a^6*b)*sign(b*x + a)/b^6